This paper is concerned with the study of the system of Laplace operators with perfect wall condition in the L_p framework. Our study includes a bounded domain, an exterior domain and a domain having noncompact boundary such as a perturbed half space. A direct application of our study is to prove the analyticity of the semigroup corresponding to the Maxwell equation of parabolic type, which appears as a linearized equation in the study of the nonstationary problem concerning the Ginzburg-Landau-Maxwell equation describing the Ginzburg-Landau model for superconductivity, the magnetohydrodynamic equation and the Navier-Stokes equation with Neumann boundary condition. And also, our theory is applicable to some solvability of the stationary problem of these nonlinear equations in the L_p framework.

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We study a kind of delayed reaction-diffusion equation with Dirichlet boundary condition. We show the existence of a sequence of values {τ_kn}_n=0,1,2,... of the parameter τ such that a Hopf bifurcation occurs when the delay passes through each value {τ_kn}. The main techniques used here are some results on nonlinear eigenvalue problems, the analysis of the characteristic equation of the linearized problem, the Liapunov-Schmidt method and the implicit function theorem.

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In this paper we give some new results on complete abstract second order differential equations of elliptic type in a Banach space. Existence, uniqueness and maximal regularity of the strict solution are proved under some natural assumptions generalizing previous theorems on the subject.

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In this paper we prove the existence of global H²-solutions and energy decay of small amplitude of the Kirchhoff type wave equation with linear localized dissipation and two types of boundary conditions in an exterior domain. Subsequently, we also consider the same problem with weakly nonlinear dissipation.

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We study the Dirichlet problem for nonlinear dissipative equations with two power (critical and sub-critical) nonlinearities of parabolic type on half lines. Taking the zero boundary conditions into consideration, we present a sufficient condition which gives sharp time asymptotics of small solutions.

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We study a generic model equation in an exterior domain. We assume the case of a non-constant coefficient function. Using a variational approach we prove existence and uniqueness theorems in anisotropically weighted Sobolev spaces. As the main tool we derive and apply an inequality of the Friedrichs-Poincaré type.

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